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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 3 - Derivatives문제 3.9.23
3장, 문제 3.9.23

Derivatives in Differential Form


In Exercises 17–28, find dy.


y = sin(5√x)

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First, identify the function y = sin(5√x). We need to find the derivative dy/dx.
Recognize that the function involves a composition of functions: the sine function and the square root function. Use the chain rule to differentiate.
The chain rule states that if you have a composite function y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). Here, f(u) = sin(u) and g(x) = 5√x.
Differentiate the outer function f(u) = sin(u) with respect to u, which gives f'(u) = cos(u).
Differentiate the inner function g(x) = 5√x with respect to x. Recall that √x can be expressed as x^(1/2), so g(x) = 5x^(1/2). The derivative g'(x) = 5 * (1/2) * x^(-1/2).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative dy/dx is found by multiplying the derivative of the outer function f with respect to the inner function g by the derivative of the inner function g with respect to x. This is essential for differentiating y = sin(5√x), where the outer function is sine and the inner function is 5√x.
추천 영상:
05:02
Intro to the Chain Rule

Derivative of Sine Function

The derivative of the sine function, sin(x), is cos(x). This basic derivative rule is crucial when applying the chain rule to differentiate y = sin(5√x). After applying the chain rule, the derivative of the outer function, sin, is needed, which transforms into cos when differentiating.
추천 영상:
03:53
Derivatives of Sine & Cosine

Derivative of Square Root Function

The derivative of the square root function, √x, is 1/(2√x). This rule is important when differentiating the inner function 5√x in the given problem. By applying this derivative rule, we can find the rate of change of the inner function, which is necessary for the application of the chain rule.
추천 영상:
01:32
Derivatives of Other Trig Functions Example 1
관련 실천
교과서 질문

In Exercises 41–58, find dy/dt.


y = ((3t − 4) / (5t + 2))⁻⁵

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교과서 질문

Second Derivatives


In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.


2√y = x – y

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교과서 질문

In Exercises 41–58, find dy/dt.


y = (1 + cos(2t))⁻⁴

257
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교과서 질문

In Exercises 43–50, find by implicit differentiation.


y² = x .

x + 1

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교과서 질문

a. Graph the function


ƒ(x) = { x, -1 ≤ x < 0

{ tan x, 0 ≤ x ≤ π/4.


b. Is ƒ continuous at x = 0?

c. Is ƒ differentiable at x = 0?


Give reasons for your answers.

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교과서 질문

Derivatives in Differential Form


In Exercises 17–28, find dy.


y = sec(x² − 1)

156
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